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Transcript

Chapter 11: Significance Testing Section 1: The Basics Definitions Significance Testing: ◦ A formal procedure for comparing observed data with a hypothesis whose truth value we want to assess. Hypothesis: ◦ A statement about a population parameter, like µ or p. Let’s think… Several cities have begun to monitor and examine paramedic response times. In Hurtville, the mean response time to all accidents involving life threatening accidents was µ=6.7 minutes. The city manager encourages them to do better each year. The following year the city manager selects a SRS of 400 life threatening calls. The average response time for this sample was 6.48 minutes. Do these data points provide good evidence that the response times have decreased since last year? Let’s think… What would we want to test? ◦ Normality? Standard deviation would be… ◦ For or Against claim? Against the claim! Types of Hypotheses Null Hypothesis, H0: ◦ The statement we are testing ◦ Statement of “no effect,” “no difference,” or no change in historical values Alternative Hypothesis, H1 or Ha: ◦ The claim about the population we are trying to find evidence for ◦ Suggests that something has changed or different than expected Looking at the paramedics… If we look back at the response times for our paramedics, what would be our null hypothesis and alternative hypothesis? ◦ H0 : ◦ Ha : ◦ This is a one-sided alternative since we are only concerned with deviations in one direction. Example 1: Larry’s car averages 26 mpg on the highway. He switches to a brand of new motor oil that is advertised to increase gas mileage. After driving 3000 highway miles with the new oil, he wants to determine if the average gas mileage has increased. What are the appropriate null and alternative hypotheses? Be sure to use the appropriate parameters and define them. ◦ µ = mean gas mileage for Larry’s car on the highway H0 : Ha : Example 2: A May 2005 Gallup Poll report on a national survey of 1208 teens revealed 72% of teens said they rarely or never argue with their friends.You wonder whether this would be true in your school so you conduct your own survey of random sample of students in your school. What are the appropriate null and alternative hypotheses? Be sure to use the appropriate parameters and define them. ◦ p = the proportion of teens in your school who rarely argue with their friends H0 : Ha : Test Statistics Compares the value of the parameter as stated in the null hypothesis with an estimate of the parameter from the sample data. Values of the estimate far from the parameter value in the direction specified by the alternate hypothesis give evidence against the null. Test Statistics To assess how far from the parameter, we standardize the estimate: Looking at the paramedics… Given the paramedic example where the average for the sample of 400 calls was 6.48, find the test statistic we would want to use to test our hypotheses. H0 : Ha : P-values The probability, assuming H0 is true, that the observed outcome would take a value as extreme as or more extreme than actually observed. The smaller the p-value, the stronger the evidence is against H0 provided by the data (ie observed result is unlikely to occur when H0 is true). Larger p-values fail to give evidence against H0. Looking at the paramedics… H0 : Ha : What is the p-value associated with testing our null hypothesis? ◦ Using our test statistic: ◦ The p-value would be: So…are we for or against our H0 ? ◦ Small p-value provides strong evidence against H0 Example 3: Does the job satisfaction of assembly workers differ when their work is machine-paced rather than self-paced? Assume the standard deviation of this normal distribution is 60. If there is no difference in job satisfaction, then the mean is 0. One study chose 18 subjects at random from a group of people who assembled electronics. These workers gave a mean of 17 (that is these workers preferred the self-paced environment). Hypotheses: H0 : Test Statistic: P-Value: Ha : Statistical Significance If the p-value is as small as or smaller than the alpha, we say that the data as statistically significant at level At a 95% confidence level, Note: Significant in a statistical sense, does not mean “important.” It just means simply not likely to happen just by chance. Statistical Significance If the P = .03 under a 95% confidence level, then So is the result statistically significant? ◦ Yes What about for ◦ No Interpretations So…we find the p-value… and we know if it’s statistically significant…now what… We want to reject H0 or fail to reject H0 ◦ If p-value > then we fail to reject H0 ◦ If p-value < then we reject H0 Steps for Significance Tests 1. Hypothesis: Identify the population and 2. Conditions: Choose the appropriate inference 3. Calculations: Find test statistic and p-value 4. Interpret: Decide statistical significance and parameter you want to draw conclusion about. State the hypothesis procedure.Verify conditions. write conclusion. Alternative Hypothesis, Ha Fail to reject H0 : ◦ We cannot find evidence to suggest Ha is true. ◦ Thus for 2 tail, we cannot find convincing evidence that the mean differs ◦ Only means the data are consistent with H0, not that we have evidence that H0 is true. ◦ Want to make statement that “since the obtained p-value is so large, we don’t have strong evidence against H0 ”